127k views
0 votes
A tunnel is shaped in the form of a semi-ellipse. The width of the tunnel is 20 feet and the height of the tunnel is 12 feet. A train is 10 feet wide and centered in the tunnel. Determine whether a 10-foot high train would have clearance to pass through. If so, by how much? If not, by how much? (Nearest inch.)

1 Answer

7 votes
Since the tunnel is in the shape of a semi-ellipse, we can use the formula for the equation of a semi-ellipse:

(x^2 / a^2) + (y^2 / b^2) = 1

where "a" is the horizontal radius (half of the width) and "b" is the vertical radius (half of the height).

In this case, we have:

a = 20/2 = 10 feet
b = 12/2 = 6 feet

We can assume that the train is centered in the tunnel, so we need to find the height of the semi-ellipse at the center (i.e., the value of "y" when "x" is 0).

Plugging in the values for "a" and "b", we get:

(0^2 / 10^2) + (y^2 / 6^2) = 1

Simplifying, we get:

y^2 / 36 = 1

y^2 = 36

y = ±6 feet

Therefore, the height of the semi-ellipse at the center is 6 feet.

To determine whether a 10-foot high train would have clearance to pass through, we need to check whether the height of the semi-ellipse at the sides is greater than or equal to 10 feet.

Plugging in the values for "a" and "b" and solving for "y" when "x" is 5 feet (half of the train's width), we get:

(5^2 / 10^2) + (y^2 / 6^2) = 1

Simplifying, we get:

y^2 / 36 = 0.75

y^2 = 27

y ≈ ±5.2 feet

Since the height of the semi-ellipse at the sides is about 5.2 feet, a 10-foot high train would not have clearance to pass through. The clearance is less than 5.2 - 5 = 0.2 feet (or about 2.4 inches).

Therefore, the train would not fit through the tunnel with 10 feet of clearance.
User Wheeliez
by
7.9k points